L. K. Waldman

Passive material properties of intact ventricular myocardium determined from a cylindrical model

The equatorial region of the canine left ventricle was modeled as a thick-walled cylinder consisting of an incompressible hyperelastic material with homogeneous exponential properties. The anisotropic properties of the passive myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varied linearly across the wall. Simultaneous inflation, extension, and torsion were applied to the cylinder to produce epicardial strains that were measured previously in the potassium-arrested dog heart. Residual stress in the unloaded state was included by considering the stress-free configuration to be a warped cylindrical arc. In the special case of isotropic material properties, torsion and residual stress both significantly reduced the high circumferential stress peaks predicted at the endocardium by previous models. However, a resultant axial force and moment were necessary to cause the observed epicardial deformations. Therefore, the anisotropic material parameters were found that minimized these resultants and allowed the prescribed displacements to occur subject to the known ventricular pressure loads. The global minimum solution of this parameter optimization problem indicated that the stiffness of passive myocardium (defined for a 20 percent equibiaxial extension) would be 2.4 to 6.6 times greater in the fiber direction than in the transverse plane for a broad range of assumed fiber angle distributions and residual stresses. This agrees with the results of biaxial tissue testing. The predicted transmural distributions of fiber stress were relatively flat with slight peaks in the subepicardium, and the fiber strain profiles agreed closely with experimentally observed sarcomere length distributions. The results indicate that torsion, residual stress and material anisotropy associated with the fiber architecture all can act to reduce endocardial stress gradients in the passive left ventricle.

A three-dimensional finite element method for large elastic deformations of ventricular myocardium : II-prolate spheroidal coordinates

A three-dimensional finite element method for nonlinear finite elasticity is presented using prolate spheroidal coordinates. For a thick-walled ellipsoidal model of passive anisotropic left ventricle, a high-order (cubic Hermite) mesh with 3 elements gave accurate continuous stresses and strains, with a 69 percent savings in degrees of freedom (dof) versus a 70-element standard low-order model. A custom mixed-order model offered 55 percent savings in dof and 39 percent savings in solution time compared with the low-order model. A nonsymmetric 3D model of the passive canine LV was solved using 16 high-order elements. Continuous nonhomogeneous stresses and strains were obtained within 1 hour on a laboratory workstation, with an estimated solution time of less than 4 hours to model end-systole. This method represents the first practical opportunity to solve large-scale anatomically detailed models for cardiac stress analysis.

A Three-Dimensional Finite Element Method for Large Elastic Deformations of Ventricular Myocardium: I—Cylindrical and Spherical Polar Coordinates

A three-dimensional Galerkin finite element method was developed for large deformations of ventricular myocardium and other incompressible, nonlinear elastic, anisotropic materials. Cylindrical and spherical elements were used to solve axisymmetric problems with r.m.s. errors typically less than 2 percent. Isochoric interpolation and pressure boundary constraint equations enhanced low-order curvilinear elements under special circumstances (69 percent savings in degrees of freedom, 78 percent savings in solution time for inflation of a thick-walled cylinder). Generalized tensor products of linear Lagrange and cubic Hermite polynomials permitted custom elements with improved performance, including 52 percent savings in degrees of freedom and 66 percent savings in solution time for compression of a circular disk. Such computational efficiencies become significant for large scale problems such as modeling the heart.

Scar remodeling and transmural deformation after infarction in the pig.

Changes in stress and tissue material properties have been proposed as important mechanical factors that may influence infarct expansion and subsequent healing. Because such changes will be reflected by alterations in the finite deformation of the tissue, we examined the direction and magnitude of myocardial deformation after coronary ligation in the pig. Methods and ResultsGold beads were implanted in the left ventricular free wall of five pigs. After ligation of the coronary supply to the region containing the markers, we used biplane cineradiography to reconstruct the three-dimensional deformations of the myocardium during single cardiac cycles as well as the remodeling deformations that occurred over time. Deformations were studied at 1 and 3 weeks after infarction. The analysis of single cardiac cycles revealed permanent loss of systolic shortening immediately after ligation. However, significant passive systolic wall thickening (P < .001) and large shears were observed at 3 weeks in regions composed almost entirely of collagen. The analysis of remodeling deformations at 1 week revealed infarct expansion with a predominant axis that varied widely. At 3 weeks, a 30% to 60% reduction in local tissue volume was measured in the infarct region, with the principal direction of scar shrinkage nearly circumferential in all animals (range, −2° to 35°). ConclusionsWe conclude that infarct expansion and scar shrinkage may be controlled by different factors. In addition, we conclude that measurement of systolic wall thickening alone is not always adequate to assess postinfarction regional contractile function.

Nonhomogeneous analysis of epicardial strain distributions during acute myocardial ischemia in the dog

To study the nonuniform mechanical function that occurs in normal and ischemic ventricular myocardium, a new method has been developed and validated. An array of 25 lead markers (approximately 4 x 4 cm) was sewn onto the epicardium of the anterior free wall of the left ventricle in an open-chest, anesthetized canine preparation. Biplane cineradiography was used to track marker positions throughout the cardiac cycle before and during episodes of acute ischemia induced by occlusion of the left anterior descending coronary artery. To estimate two-dimensional nonhomogeneous deformations in the region at risk and its border zone with normally perfused tissue, surfaces defined by bicubic Hermite isoparametric finite element interpolation were fitted by least squares to the three-dimensional marker coordinates in successive cine frames. Global smoothing functions prevented ill-conditioning in areas of low marker density. Continuous distributions of systolic finite strains referred to the end-diastolic state were obtained under normal and ischemic conditions without the conventional assumption of homogeneous strain analysis. Substantial regional variations in epicardial strains were observed in both the normal and ischemic heart. The method was validated in regions of small to moderate strain variations by comparing the continuous distributions of strain components with piecewise-constant measurements made using marker triplets and homogeneous strain theory. The influence of marker density was examined by recomputing strains from surfaces fitted to subsets of the original array. Further validation of moderate to large strain variations was obtained by simulating a nonuniform distribution of stretch across a planar sheet and computing strains both analytically and using the current method. The new method allows for more comprehensive measurements of distributed ventricular function, providing a tool with which to quantify better the nonhomogeneous function associated with regional ischemia.

Ventricular wall motion

Before discussing Dr. Elzinga’s work, I thought it might be appropriate to take a moment to ease the transition from the first chapters on the mechanics of the isolated myocardium to ventricular mechanics and describe briefly the relationship between forces and deformation in the wall of the ventricle. To translate the elegant studies on isolated muscle discussed in the previous chapters to the intact heart, one must find a way to measure force and deformation in the complex thick wall of the ventricle. Most investigators have used membrane theory to calculate wall-forces and have measured deformation in the wall with some type of uniaxial measurement. As I will attempt to show you in recent data from our laboratory, motion in the ventricular wall is probably much too complex to allow either of these two analyses.

Effect of coronary artery reperfusion on transmural myocardial remodeling in dogs.

BACKGROUND The effects of reperfusion after coronary occlusion on transmural remodeling of the ischemic region early and late after nontransmural infarction must importantly affect the recovery of regional function. Accordingly, analysis of local volume and three-dimensional strain was performed using a finite element method to determine regional remodeling. Systolic and remodeling strains were measured using radiographic imaging of three columns (approximately 1 cm apart) of four to six gold beads implanted across the left ventricular posterior wall in 6 dogs. METHODS AND RESULTS After a control study, infarction was produced by 2 to 4 hours of proximal left circumflex coronary artery occlusion followed by reperfusion. Follow-up studies were performed at 2 days, 3 weeks, and 12 weeks with the dogs under anesthesia and in closed-chest conditions. Biplane cineradiography was performed to obtain the three-dimensional coordinates of the beads. At 2 days, end-systolic strains were akinetic with loss of normal transmural gradients of shortening and thickening. Remodeling strains (RS) were determined by use of a nonhomogeneous finite element method by referring the end-diastolic configuration during follow-up studies to its control state at matched end-diastolic pressures and heart rates. Tissue volume at 2 days increased substantially, more at the endocardium (30 +/- 7%) than at the epicardium (5 +/- 12%, P < .01); the increase was associated with an average RS in the wall-thickening direction of 0.18 +/- 0.15 (P < .01) with all other RS near zero. At 12 weeks systolic function partially recovered, with normal wall thickening in the epicardium (radial strain, 0.081 +/- 0.056 [control] versus 0.113 +/- 0.088 [12 weeks]) but with dysfunction in the endocardium (0.245 +/- 0.108 [control] versus 0.111 +/- 0.074 [P < .01] [12 weeks]). This inability of the inner wall to recover function may be related to increased transmural torsional shear and negative longitudinal-radial transverse shear in the inner wall. Volume loss occurred at 12 weeks in the endocardium (-36 +/- 16%) corresponding to transmural gradients in longitudinal RS and both transverse shear RS. Negative longitudinal RS was greater at the endocardium (-0.20 +/- 0.10) than at the epicardium (-0.06 +/- 0.05, P < .01). CONCLUSIONS These results indicate the presence of marked subendocardial edema 2 days after reperfusion following 2 to 4 hours of coronary occlusion. At 3 months after reperfusion, however, there was volume loss in the inner wall due to shrinkage along the myofiber direction with reduced transmural function and loss of longitudinal shortening, while both tissue volume and function recovered completely in the outer wall.

Regional myocardial perfusion and mechanics: A model-based method of analysis

AbstractA new parametric model-based method has been developed that allows epicardial strain distributions to be computed on the left ventricular free wall in normal and ischemic myocardium and integrated with the regional distributions of anatomic and physiological measurements so that underlying relationships can be explored. An array of radiopaque markers was sewn on the anterior wall of the left ventricle (LV) in three anesthetized open-chest canines, and their positions were recorded using biplane video fluoroscopy before and 2 min after occlusion of the left anterior descending coronary artery. The three-dimensional (3D) anatomy of the LV and epicardial fiber angles were measured post-mortem using a 3D probe. A prolate spheroidal finite element model was fitted to the epicardial surface points (with <0.2 mm accuracy) and fiber angles (<5° error). Regional myocardial blood flows (MBFs) were measured using fluorescent microspheres and fitted into the model(<0.3 ml min−1 g−1 error). Epicardial fiber and cross-fiber strain distributions were computed by allowing the model to deform from end-diastole to end-systole according to the recorded motion of the surface markers. Systolic fiber strain varied from −0.05 to 0.01 within the region of the markers during baseline, and regional MBF varied from 1.5 to 2.0 min−1 g−1. During 2 min ischemia, regional MBF was less than 0.3 min−1 g−1 in the ischemic region and 1.0 ml min−1 g−1 in the nonischemic region, and fiber strain ranged from 0.05 in the central ischemic zone to −0.025 in the remote nonischemic tissue. This analysis revealed a zone of impaired fiber shortening extending into the normally perfused myocardium that was significantly wider at the base than the apex. A validation analysis showed that a regularizing function can be optimized to minimize both fitting errors and numerical oscillations in the computed strain fields.

Distributed mechanics of the canine right ventricle: Effects of varying preload☆

Fundamental questions in the mechanics of the right ventricle (RV) include: what are the distributions of diastolic and systolic strains across the RV epicardium and how do these strains change with increasing preload? Arrays (approximately 4 x 4 cm) of 25 to 30 lead markers were sutured to the epicardium of the RV anterior free wall in 6 open-chest, anesthetized dogs. Biplane cinéradiography (16 mm, 120 fps) was used to track marker positions throughout the cardiac cycle as loading conditions were altered by intravenous volume infusion. Continuous two-dimensional nonhomogeneous deformations were estimated across the region by fitting high-order finite element surfaces to the three-dimensional marker coordinates in successive ciné frames. End-systolic strains referred to end-diastole did not change with increasing preload, but did exhibit considerable longitudinal variation, e.g. the principal strain associated with maximal shortening (E1) was more than twice as great nearer the apex (E1 = -0.18 +/- 0.08) than in more basal (E1 = -0.09 +/- 0.05) regions. However, large amounts of lengthening occurred during diastolic inflation. End-diastolic extensional strains referred to an unloaded configuration were moderate at low pressure (E2 = 0.13 +/- 0.08) but increased to large values at high preloads (E2 = 0.28 +/- 0.11). End-diastolic strains also showed considerable longitudinal variation, i.e. near the base lengthening (E2 = 0.31 +/- 0.13) tended to be much greater than near the apex (E2 = 0.15 +/- 0.12). These results indicate that both diastolic sarcomere lengths and systolic sarcomere shortening increase in proportion to diastolic loading leaving end-systolic sarcomere strains unchanged.

Nonhomogeneous Ventricular Wall Strain: Analysis of Errors and Accuracy

Nonhomogeneous distributions of strains are simulated and utilized to determine two potential errors in the measurement of cardiac strains. First, the error associated with the use of single-plane imaging of myocardial markers is examined. We found that this error ranges from small to large values depending on the assumed variation in stretch. If variations in stretch are not accompanied by substantial regional changes in ventricular radius, the associated error tends to be quite small. However, if the nonuniform stretch field is a result of substantial variations in local curvature from their reference values, large errors in stretch and strain occur. For canine hearts with circumferential radii of 2 to 4 cm, these errors in stretch may be as great as 30 percent or more. Moreover, gradients in stretch may be over- or underestimated by as much as 100 percent. In the second part of this analysis, the influence of random measurement errors in the coordinate positions of markers on strains computed from them is studied. Arrays of markers covering about 16 cm2 of ventricular epicardium are assumed and nonuniform stretches imposed. The reference and deformed positions of the markers are perturbed with Gaussian noise with a standard deviation of 0.1 mm, and then strains are computed using either homogeneous strain theory or a nonhomogeneous finite element method. For the strain distributions prescribed, it is found that the finite element method reduces the error resulting from noise by about 50 percent over most of the region. Accurate measurements of cardiac strain distributions are needed for correlation with and validation of realistic three-dimensional stress analyses of the heart.(ABSTRACT TRUNCATED AT 250 WORDS)